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304 PRICE At the end of the first period, a rf pulse is applied that has the effect of reversing the sign of the precession Ž i.e., the sign of the phase angle. or, equivalently, the sign of the applied gradients and static field. At time t1 , a second gradient pulse of equal magnitude and duration is applied Žn.b., the pulse has the effect of changing the sign of the first gradient pulse; this leads to the idea of an ‘‘effective’’ field gradient; see The Macroscopic Approach . . If the spins have not undergone any translational motion with respect to the z-axis, the effects of the two applied gradi-
ent pulses cancel and all spins refocus. However, if the spins have moved, the degree of dephasing due to the applied gradient is proportional to the displacement in the direction of the gradient Ži.e., the z-direction. in the period Ži.e., the duration between the leading edges of the gradient pulses . . Thus, at the end of the echo sequence, the total phase shift of spin i relative to being located at z 0 is given by ½ H 5 i 0 t1 t1 i first period t1 B g z Ž t. ½ 0 H i dt5 t1 second period t1t1 g z Ž t. dt z Ž t. ½H i H i dt5 t1 t1 Ž 2. B g z Ž t. dt 9 We should recall that in NMR we are concerned Ž with an ensemble of nuclei with different spatial PULSED-FIELD GRADIENT NMR 305 starting and finishing positions . , and thus, the normalized intensity Ž i.e., an attenuation. of the echo signal at t 2 Ž 58, 60 . , i.e., SŽ 2. is given by H Ž . Ž . Ž . i g0 S 2 S 2 P ,2 e d. 10 where SŽ 2. is the signal Ž g0 i.e., resultant magnetic moment. in the absence of a field gradient. If we consider only the real component of SŽ 2 . , and recalling De Moivre’s theorem, we have i e cos i sin 11 H SŽ 2. SŽ 2. PŽ ,2. cos d 12 g0 where PŽ ,2. is the Ž relative. phase-distribution function. For the PFG sequence, some authors write PŽ , . , since it is the gradient pulses and the separation between them that constitute the Figure 2 A schematic representation of how the Stejskal and Tanner Ž or PFG. pulse sequence measures diffusion and flow. This is a Hahn spin-echo pulse sequence with a rectangular gradient pulse of duration and magnitude g inserted into each delay. The separation between the leading edges of the gradient pulses is denoted by . The applied gradient is generally along the z-axis Ž the direction of the static field . . The second half of the echo is digitized Ž denoted by dots. and used as the free induction decay Ž FID . . In this schematic description, we assume that we start the pulse sequence with a sample consisting of four in-phase spins Ž really an ensemble! . and we consider only the precession due to the gradient Ž i.e., we use a rotating reference frame rotating at . 0 . We assume that the center of the gradient coincides with the center of the sample Ž i.e., z 0 . . Accordingly, the spins above and below this point acquire phase shifts owing to the gradient pulses, but in opposite senses. In the absence of diffusion, the effect of the first gradient pulse, denoted by the curved arrows in the first phase diagram, is to create a magnetization helix Ži.e., the solid ellipses in the center phase diagram. with a pitch of 2 Ž g . . Although we have represented the gradient pulses as having a finite width it is easier to consider them in the limit of 0 Ž i.e., the short gradient pulse limit . . The pulse reverses the sign of the phase angle Ž i.e., the dotted ellipses in the center phase diagram . , and thus, after the second gradient pulse, the helix is unwound and all spins are in phase, which gives a maximum echo signal. In the presence of diffusion, the winding and unwinding of the helix are scrambled by the diffusion process, resulting in a distribution of phases, although it is not easily seen since our sample consists of only four spins. Larger diffusion would be reflected by poorer refocusing of the spins, and consequently by a smaller echo signal. The effects of restriction upon the diffusion process will also contribute to this loss of phase coherence. In the absence of any background gradients, diffusion in the periods before Ž e.g., 0 t . 1 and after Ž i.e., t . 1 the gradient pulses does not affect the signal attenuation. In the presence of flow Ž imagine that the outflowing spins are replaced by inflowing spins. along the direction of the gradient Ž in the z direction with velocity in the present example. and neglecting diffusion, all the spins receive the same change in phase. The greater the flow is, the larger is the net phase change. If both diffusion and flow processes are present, then the whole diffusion-induced phase distribution receives a net phase shift.
Page 1 and 2: Pulsed-Field Gradient Nuclear Magne
Page 3 and 4: PULSED-FIELD GRADIENT NMR 301 Figur
Page 5: the gradient oriented along the z-a
Page 9 and 10: ent echo, on the other hand, refocu
Page 11 and 12: For the case of Ž three-dimensiona
Page 15 and 16: and S corresponds to the value of
Page 17 and 18: every time a pulse is applied. Thu
Page 19 and 20: Finally, noting the standard integr
Page 21 and 22: Now, if we integrate Eq. 83 over al
Page 23 and 24: In the long time limit 1 Ž i.e.,
Page 25 and 26: In the next section, we will illust
Page 29 and 30: sor, D app. When a single effectiv
Page 33 and 34: Experimental aspects of PFG NMR wil
Page 35 and 36: 32. P. C. M. Van Zijl and C. T. W.
Page 37 and 38: 99. A. Caprihan, L. Z. Wang, and E.

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